Mechanical differentiator for smoothing target tracking data



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"m "aw-MM* Wu 29476925@ l R. B. BLACKMAN MECHANICAL DIFFERENTIATOR FOR SMOOTHING TARGET TRACKING DATA Filed Allg. 20, 1945 ammi.;

.Fuy i9, 1949.

@MC/WAN R. B. BLACKMAN MECHANICAL DIFFERENTIATOR FOR SMOOTHING TARGET TRACKING DATA July g, Filed Aug. 2O

Patented July 19, 1949 MECHANICAL DIFFERENTIATOR FOR SMOOTHIN G TARGET TRACKING DATA Ralph B. Blackman, Cranford, N. J., assigner to Bell Telephone Laboratories, Incorporated, New York, N. Y., a corporation of New York Application August 20, 1945, Serial No. 611,639

(Cl. 23S-61.5)

4 Claims.

This invention relates to an improvement in mechanical differentiators, particularly useful in a mechanical system for indicating an angular velocity.

The general object of the invention is to provide improved means for indicating the angular velocity of a shaft of which the varying angular position and the angular velocity represent respectively the varying value of an observed quantity and the time rate of change thereof.

The observed quantity may be the horizontal displacement of an airplane in iiight of which the position is continuously recorded by an observer on the ground. The observers instruments may themselves be iiawless but his use of them is usually affected by random errors of overrunning or underrunning and when a shaft is rotated to follow the value of the observed quantity, the angular position of this shaft does not change uniformly with time even if the observed quantity is so changing. It thus becomes necessary to provide means for measuring the shaft angular velocity and smoothing out the random variations therein.

A further object of the invention is therefore to provide improved means for obtaining a smoothed average of the angular velocity of the rotating shaft by elimination of random errors of observation.

It may be shown by mathematical analysis that if a series of observations of instantaneous angular velocity are averaged over a convenient time interval, the angular velocity being found constant if perfectly observed, this average should be obtained by parabolic weighting of the successive angular velocities; that is, the least weight should be given to the velocities observed at the beginning and at the end, and the maximum weight to those at the center, of the interval. The average velocity so obtained will be identical with that derived from the slope of a straight line tted by a least squares adjustment to the successive angular shaft positions plotted against time and the random errors alluded to will be filtered out.

Accordingly, it is another object of the invention to provide means for obtaining with approximately parabolic weighting over a convenient time interval the average value over that interval of the angular velocity of a rotating shaft.

While the apparatus of the invention may be made sufciently complex to approach as closely as desired the ideal parabolic Weighting of instantaneous angular velocities, the invention will be set forth with reference to two embodiments which approximate the ideal and constitute improvements over the mechanisms known to the prior art.

The electrical analogue of the mechanical problem is found where the varying observed quantity is represented by a varying voltage which is differentiated with respect to time as, for example, by the differentiating amplifier disclosed by E. L. Norton in United States Patent 2,324,797, July 20, 1943. It is known that an amplier having a feedback path including an integrating circuit provides an output voltage which is the time derivative of the input voltage. The present invention applies this principle to mechanical differentiators, so that another object of the invention is to provide an improved mechanical transmission system of the type having between input and output a mechanical feedback path comprising mechanical integrating means.

The invention will be understood from the following description read with reference to the accompanying drawings in which:

Fig. 1 presents curves exhibiting certain weighting functions, ideal as well as practicable;

Fig. 2 shows a simple mechanical system according to the prior art for indicating the angular velocity of a rotating shaft;

Fig. 3 shows schematically one embodiment of the present invention; and

Fig. 4 similarly shows another embodiment of the invention.

In all figures, like numerals and letters denote like elements.

It will be convenient to use the notation of the operational calculus to designate the various shaft rotations involved in the apparatus shown in Figs. 2 to 4, inclusive. In this notation pf (t) indicates the time derivative, while 1 'if (i) indicates the time integral of the function f(t). No more than the results of mathematical analysis will be stated here, for the reason that a mathematical discussion is unnecessary to an understanding of the invention. The angular position of a shaft will be designated by 0, the angular velocity by 0 with subscripts where appropriate.

The mechanical transmission circuit beyond the rotating shaft of which the angular velocity is to be indicated, is analogous to an electrical transmission circuit in which the output current equals input voltage multiplied by the steady state transfer admittance, which may be chosen by circuit design to result in an output current proportional to the time derivative of the input voltage. The indicial admittance of the circuit is the weighting function according to which is averaged the output current to represent the best time derivative of the input voltage, which may be varying at a uniform rate plus the superposition of random irregularities.

To solve the problem of mechanical diierentiation, one species the desired mechanical indicial admittance, that is, a mechanical weighting function; derives from this the mechanical transfer admittance; and finally designs a mechanical system possessing approximately the transfer admittance desired. In both the electrical and the mechanical case, the ideal solution is foregone for practical reasons.

Referring now to Fig. 1, curves A, B, C and D are weighting functions plotted with weight factor as ordinate against time as abscissa, the unit of time being proportional to the time constant of the averaging mechanical network. This time constant is determined, as later explained, by the gear ratios and motor speeds in Figs. 2, 3 and 4.

In Fig. 1, curve A is the exponential weighting function given by the simple differentiator of Fig. 2. Curve B is the ideal parabolic Weighting which is zero at time zero and unity. Curve C is one approximation to curve B, practically realized by the present invention in the embodiment shown in Fig. 3 and curve D is a better approximation than curve C which is realized by the system shown in Fig. 4.

The improvements provided by the present invention are best discussed after a description of Fig. 2, which shows a simple system of apparatus whereby the angular velocity of a rotating shaft may be measured. Shaft I0, turning in the sense indicated by the arrow, occupies momentarily an angular position 01 which varies with time at the rate 01 which is to be determined. Through differential gear I'I, shaft I0 drives shaft I2 on which is carried pointer I3, the angular position of which is read on dial I4. From shaft I2 through bevel gears I5 is driven shaft I6 carrying pinion II which moves rack I8 to the right, when shaft I6 turns in the indicated direction.

A constant speed motor CSMI, of any known design, drives as indicated disc 20, which through contacting spherical balls 2| turns roller 22 about an axis at right angles to the axis of motor CSMI at a rotational speed determined by the radial location of balls 2| on disc 2D. The axis of roller 22 is prolonged as shaft 23, terminated in spur gear 24 meshing with ring gear 25 of differential II. Balls 2|, which may be of any desired even number, two being shown in Fig. 2, are by any suitable means constrained by the motion of rack I8 to move radially with respect to disc which results in an angular speed of roller 22 proportional to the angular position of shaft I6. The disc and roller system just described may be such as shown in United States Patent 1.317.915, October '7, 1919 to H. C. Ford or in United States Patent 2,002,585, May 18, 1935 to J. J. Rothwell et al. It will at once be recognized that the angular position of shaft 23 changes in any time interval by an amount proportional to that interval multiplied by the radial shift of balls 2| from the center of disc 20; that is, to the rotation of shaft I5, which is the same as that of shaft I2 if gears I5 are of unity ratio. When shafts I2 and I6 through pinion I1 and rack I8 are set to center balls 2li on disc 20, pointer I3 is adjusted on shaft I2 to read zero on scale I4.

Now, if the angular position 92 of shaft I2 represents an angular velocity, the rotation of shaft 23 during any time interval is proportional to the time integral of 62 over that interval. In terms of the operational calculus, the angular position 03 of shaft 23 is By suitable choice of the gear ratio between gears 24 and 25 a fraction of this integral may be impressed as a rotation of ring gear 25 just sufficient to offset the rotation of shaft I0, whereupon shaft I2 comes to rest at an angular departure 02 from the zero setting above defined. When this cornes about no further rotation of shaft I2 takes place. 02 read on scale I4 is then the measure of the angular velocity of shaft I0: 01=p0i=6a The elements from gears I5 through ring gear 25 constitute an integrating mechanical feedback path between output shaft I2 and input shaft I0. The regulated power supply 30, of any known form, supplies power to drive motor I9, the speed of which may be adjusted by varying resistor 3|. The integrator above described is generally designated as included in dashed rectangle I, Fie. 2.

Whenever a sudden change is brought in the angular velocity of shaft I0 a certain time interval must elapse before shaft I2 turns to the corresponding new steady reading on scale I 4. The readjustment of shaft I2 is at first rapid and then more slow. As may be shown by an analysis here omitted, this readjustment obeys the law of curve A of Fig. 1, the exponential weighting function which attaches greatest weight to the earliest information.

Fig, 2 shows also how the angular velocity 02=p01 may be shown electrically as well as mechanically. Shaft I2 is prolonged to carry brush 32 traversing linear potentiometer 33 supplied with voltage from battery 34, which may be included in power supply 30. Variable resistance 36 may be adjusted to control the scale factor at which voltrneter 35, connected between brush 32 and the grounded end of potentiometer 33, indicates a voltage proportional to 02.

A change in velocity of shaft I0 occurs when that shaft is first connected with the apparatus with which an observer is tracking an airplane, for example, and other velocity changes occur when shaft Ill is advanced or retarded by the observer in correcting for overrun or underrun in his observations. In such tracking, the earlier corrections are usually larger than the later, and the earlier of the velocity variations averaged by the system of Fig. 1 are given undue weight by exponential weighting. It is therefore desirable to provide a weighting function approximating the ideal curve B of Fig. 1.

This is done by an elaboration of the simple system, in which again are used the elements included in dashed rectangle I of Fig. 2.

To derive a physically realizable approximation to the parabola of Fig. 1, consider the ideal indicial admittance Ao t and the correspondtransfer admittance Yom), where p is the differential operator Ao(t) =6t(1t) for t between 0 and 1 =0 for t 0 or 1,

'I5 where the theoretical response time of the mechanical circuit is unity. Ao(t) is drawn as curve B of Fig. 1.

The corresponding transfer admittance may be shown to be given by The specic embodiments shown in Figs. 3 and 4 constitute approximations involving the righthand member of Equation 2 to the second and to the fourth power of p, respectively.

Y1(1U) :gg 1

which may be written as curve C.

As another approximation, let

then

1 18 3 4l G1=a b 1 Dd so that Yup) :i12 p+70 (6) p 41 p 41 p 41 which may be written 41 (p4-4.26) @+3.36 +1357) (p+3.3c-i3.57)

leading to the indicial admittance Az(t), plotted in Fig. l as curve D:

Stated in its most general form, the relationship of the angular displacement 02 of output shaft I2 to 01, the angular displacement of shaft lll, Fig. 2, is 02=Y(p) 01 and this is analogous to the electrical circuit formula I =Y(p) E, in which I is the output current of an electrical network of transfer admittance Y(p) and E is the electromotive force impressed across the network input. By suitable choice of YW), the output current may be made proportional to either the time integral or the time derivative of the input voltage, and the same may be clone in a mechanical system. The present invention contemplates the latter named relationship between input and output shaft angular displacements.

In the simple diiferentiator shown in Fig. 2,

and the angular displacement of shaft l2 represents the angular velocity of shaft I0, exponentially averaged.

In any case, where 02(t)=Y(p).0(t), it may be shown that the response 02(t) at time t is given by the equation z ce) =f Ac-n 01 o) dn 91 Where )1 is time counted from the past from the same point of reference as t, and 0100 is the angular velocity (measured continuously at successive values of A) of the shaft having the angular displacement mt). The age at time t of the measurement made at time A is written as 1=t and z f Ac-mamdr becomes L 21(1)@ c-fwf,

where A(f) is the weight factor applied to the data of age r.

If such a weighting function as Ah) can be provided, then where i is the Weighted average of is the transfer admittance determining the final value of 62. Y p is to be mechanized.

It can be further shown that to be practically mechanized Y(p) must be of the form where M(p) and N(p) are polynomials in p with constant real coefficients. We shall consider the mechanization of the cases where NMO) is either one or two degrees higher in p than Mw).

To put the operational equation into a form suitable for mechanization, it is carried through the following transformations, in steps as follows:

where a=0 if N(p)lis only onefdegree higher in 7 p than Mo. 'rfm is at most of the same degree as Mm). Then PMQJ) is expanded into partial fractions of which one is of the form Now,

and the others are each of the form P out in common to all these terms, we have (10) 1 1 02:.1117-1- E101-glai- Y (p) -l- Y.. (p) +ete1o2] In Equation 10 the Y(p)s are each of the form The terms Ym(p), Yum), etc., are all of the form PM@ N (p) and so introduce no new problems in the design now to be proposed.

When Ai= and B1=1 and the terms in Ym(p), etc., are omitted, Equation reduces to the form P-I-b obviously involves the generation of 02 as the integral of 614-1202, that is,

and in the system of Fig'. 3 shaft I2, instead of being directly the output shaft of differential II is thereto connected through an intervening mechanical integrator of the general form included in dashed rectangle I of Fig. 2.

Referring now to Fig. 3, input shaft I0 and output shaft I2 continue to be connected by the feedback path (dashed rectangle I) entering differential gear II. The direct connection from differential gear Il to output shaft I2 is here interrupted to include, in dashed rectangle 2, the intervening integrator mentioned in the preceding paragraph. Opposite shaft Ill, output shaft 40 leads to differential gear 4I and therethrough to shaft 42 terminating in bevel gears 43. Through gears 43, shaft 42 drives shaft 44 carrying pinion 45 engaged by rack 46. Rack 46 controls, as does rack I8, an integrating element generally designated as 50 and identically like elements CSMI to 22 of Fig. 2. Output shaft 5I of integrator 50 drives, through bevel gears 52 and 53, nal output shaft I2. The last-named shaft is provided with bevel gears I5 and carries pointer I3 as in Fig. 2. The feedback path between shafts I2 and I0 is identical with that in Fig. 2.

On shaft I2, at the end opposite pointer I3, is carried spur gear 54 meshing with ringgear 55 of differential gear 4I. The angular displacement 02 of shaft I2 is here related to the angular displacement 64 of shaft 40 by the relation 1 (9a-m04 Since 04 is, as in Fig. 2 for 62, equal to the final displacement of shaft I2 in Fig. 3 is 92 p+b(0-P02) Examination of the diagram of Fig. 3 shows that while the initial effect of the rotation 04 is to bring about a proportional angular velocity of shaft 5I, the increasin-g displacement Hz of shaft I2 progressively cancels this effect so that integrator 5U ceases to operate simultaneously with the coming to rest of shaft I2, provided the angular velocity 01 is strictly constant.

When shaft I2 in Fig. 3 comes to rest, it may be shown that its angular position 02 is proportional to the angular velocity 01 of shaft I0 weighted substantially in accordance with curve C, Fig. 1. It will be observed that angular position feedback has been introduced from shaft I2 to ring gear 55, in addition to the feedback mechanical integrating path from shaft I2 to ring gear 25. The added path, inclosed in dashed outline 2, is an improvement provided by the present invention; an additional improvement is shown in Fig. 4.

Without detailing the necessary mathematical analysis, it may be stated that the approximation to the parabolic Weighting function, curve B of Fig. l, may be progressively improved by adding any number of admittance paths in parallel with the path between gears I5 and integrator of Fig. 2. Each of these added admittance paths requires the introduction of a differential gear to enable its output to join that from gears I5 to be operated on by integrator 90. It will suiiice here to illustrate the addition of one such admittance.

Referring now to Fig. 4, a system providing a weighting function substantially that shown by curve D, Fig. 1, the system of Fig. 3 is elaborated by adding a third integrator included in dashed rectangle 3. Bevel gears 60 transmit the motion of shaft I2 to auxiliary shaft 6 I, from which bevel tiitiiiii NUUW gears 62' redirect the motion to enter differential gear 63. The output shaft 64 of gear 63 carries bevel gears 65 whereby is driven rack 68 controlling the integrator generally designated as 10. of like form to the previously described integrators in Figs. 2 and 3. Through bevel gears I2 and 13, and ring gear 'I4 the output of integrator 'I0 introduces into differential 63 a motion proportional to the time integral of the motion of shaft G4 and in a sense adapted to check the motion of the last named shaft.

To add the motion of shaft 64 to that of shaft I6, differential is introduced between gears I5 and pinion Il and rack I8 is moved from above, as in Fig. 2, to below pinion I'I. Shaft 64 is prolonged to drive through bevel gears 16, ring gear 'Il of differential 'I5 in the sense to add the motion of shaft 64 to that of shaft I6', the latter driven through differential 'I5 by shaft I6. The combined motion of shafts 64 and I6' is then integrated by integrator 9D, as previously described in connection with Fig. 2, to cancel progressively the motion of the output shaft of differential Il.

In Figs. 2, 3 and 4, full line arrows indicate the nal directions of movement of the racks in integrators 90, 50 and l0, when shaft I2 is moving from zero to its final rest position, as indicated by the dashed arrows.

The racks of integrators 50 and 'I0 initially move as shown by the dashed arrows and then, as shaft I2 approaches its nal position, return (full line arrows) to center the transmitting balls on their driving discs if b1 is constant. The rack of integrator S0 moves only as shown by the full arrow, coming to a final position as shaft I2 does so. It will be noted that the directions of rotation of the discs driven by constant speed motors in all three integrators are the same. It is to be understood that this is only illustrative, and `that the motor rotations may be conveniently chosen if regard is had to preserve the sense of the shaft motions as indicated. Further, a little consideration of Fig. 4 will show that as for integrator 50 in Fig. 3, the coming to rest of shaft I2 will be accompanied by the cessation of integrating operation by both integrators 50 and 'III if the motion 01 of shaft IU is strictly uniform with time. Only integrator 90 will in that event continue to operate to make 02=pi=01. Since the constant speed motors run continuously, all transmitting balls come to center when shaft I0 is at rest.

The time constant of the system is determined by the various gear ratios and integrator dimensions, together with the speeds of the constant speed motors CSMI, CSM2 and CSM3, and may be chosen at pleasure by adjusting these speeds. It is :to be understood that all the constant speed motors may be supplied, by means shown only for CSMI, from power supply 30, Fig. 2. Since the time constant may be thus simply controlled, it is deemed unnecessary herein to prescribe gear ratios and dimensions of parts; these may be chosen as seems convenient to the designer.

Adjacent the important shafts in Figs. 2, 3 and 4 are illustrated their respective angular displacements as 0i, 02, etc. It may be profitable to tabulate the relations of these displacements in the several figures, stating the corresponding Weighting functions.

In Fig. 2:

a P 02-91-5-92 m 91 weighting function exponential, curve A of Fig. 1.

InFig. 32

Equation 3, curve C of Fig. 1.

In Fig. 4:

l @gmt weighting function approximately parabolic, Equation 6, curve D of Fig. 1.

In the above expressions for 02, the constants a, b and c refer of course to the gear ratios concerned in driving the ring gears.

Obvious modifications in the graduation of scale I4 and in the battery connection to potentiometer 33 can be made to enable the system to indicate angular velocities of shaft II) in the opposite sense to that shown in the drawings.

Calibration of the system is made by imposing, from any suitable source a known constant angular velocity on shaft I0, the transmitting balls of the several integrators 90, 5D and I0 being 5 centered on their respective driving discs which are in rotation at constant speed. The final position of pointer I3, and the time taken to reach that position, are noted. The disc speeds may be changed if this time is too long or too short, and provision, not shown, may be made to gear up or down the movement of pointer I3. It is obvious from a consideration of Fig. 2, for example, that the higher the speed of disc 20, the less the displacement of shaft I 2 for a, given angular velocity 5 of shaft I0. If the response time is to be shortened and the scale factor on scale I4 is to be kept the same, pointer I3 must be geared up in the same ratio as the disc speed is increased.

The choice among the systems of Figs. 2, 3 and 4 60 is to be based on the character of the motion 01 of shaft I0. If this motion is uniform with time, the simple system of Fig. 2 will serve. If 01 is a motion uniform with time plus random increases and decreases in velocity, the system of Fig. 4 55 is preferred to that of Fig. 3 unless the additional mechanism is unwelcome. The apparatus of Fig. 4, realizing curve D of Fig. 1, more closely follows the parabola B, and thus provides a more accurate result than does the apparatus of Fig. 3 70 which corresponds to curve C. Moreover, since curve D reaches zero sooner than curve C, the error due to the curve area beyond t=1 is less.

It will be understood that the specific integrating elements shown in integrators 90, 50 and 10 75 are illustrative only, there being numerous mechanical speed changing systems which may be used, in place of those shown, to produce the same result. In Fig. 2, for example, all that is required is a means for conferring on shaft 23 an angular velocity proportional to the angular displacement of shaft I2 from e, zero position, and any means for so doing may replace that shown without departing from the present invention.

For a shaft Ill turning with a gradually varying angular velocity, with or without random irregularities, the invention provides, in the embodiments shown in Figs. 3 and 4, an angular velocity averaged with substantially parabolic weighting over a time interval of selectable length. The speed of the timing discs in the integrators should be set for a response time several times the period of the principal irregularities. The system of the invention then provides a running average angular velocity of the input shaft, smoothing out random fluctuations.

I am aware that mechanical differentiating and smoothing networks, combining differential gears and mechanical integrators, are known. I-Iowever, the design herein disclosed and based upon the derivation of Equation 10 is believed to include advantages not heretofore realized in such mechanisms. These advantages are:

1. The design is independent of the number of real or conjugate, or both real and conjugate, complex pairs of exponential functions of time which are combined to obtain a satisfactory weighting function.

2. The number of precision parts is also indi.- pendent of the number of exponential functions.

Since if the angular velocity of shaft I0 is absolutely constant all integrators except that generally designated as 90 eventually return to a neutral position, only integrator 90 needs to be of high precision. In addition, all gears may be of low precision, except, as in Fig. 4, gears I5, differential 15, pinion I1 and rack I8, and precise speed regulation is required only for motor CSMI.

What is claimed is:

1. Mechanism for indicating the angular velocity of a first rotating shaft comprising a second shaft, means including a first differential gearing for driving the second shaft from the first shaft and in the opposite sense of rotation, a third shaft, means including a second differential gearing for driving the third shaft from the second shaft in the sense of rotation of the first shaft, a fourth shaft, means including a mechanical integrator for driving the fourth shaft from the third shaft in the sense of rotation of the second shaft at an angular velocity proportional to the angular displacement of the third shaft, means for introducing from the fourth shaft to the third shaft angular displacement feedback proportional to and in the sense of the angular displacement of the fourth shaft, means including a mechanical integrator connected between the fourth shaft and the rst dilferential gearing for adding to the angular velocity of the second shaft an opposite angular velocity proportional to the angular displacement of the fourth shaft, and means for indicating the last-named angular displacement.

2. A mechanical system controlled by the rotation of a first shaft comprising a second shaft driven through differential gearing from the rst shaft, a third shaft driven through differential gearing from the second `shaft, a fourth shaft, a mechanical path including integrating means between the third shaft and the fourth shaft, means for introducing angular position feedback from the fourth shaft to the second-named differential gearing and a mechanical feedback path including integrating means between the fourth shaft and the first-named differential gearing, whereby rotation of the first shaft occasions an angular displacement of the fourth shaft proportional to the angular velocity of the first shaft.

3. Mechanism for indicating the angular velocity of a first rotating shaft comprising a second shaft, means including a rst differential gearing for driving the second shaft from the first shaft and in the opposite sense of rotation, a third shaft, means including a second differential gearing for driving the third shaft from the second shaft in the sense of rotation of the first shaft, a fourth shaft, means including a mechanical integrator for driving the fourth shaft from the third shaft in the sense of rotation of the second shaft at an angular velocity proportional to the angular displacement of the third shaft, means for introducing from the fourth shaft to the third shaft angular displacement feedback proportional to and in the sense of the angular displacement of the fourth shaft, a fifth shaft, means including a third differential gearing for driving the fth shaft from the fourth shaft in the opposite sense of rotation, means including a mechanical integrator connected between the fifth shaft and the third differential gearing for adding to the motion of the fifth shaft an angular velocity proportional and opposed to the angular displacement of the fifth shaft, a sixth shaft, means including a fourth differential gearing for imparting to the sixth shaft an angular displacement proportional to the sum of the angular displacements of the fourth and fifth shafts, means including a mechanical integrator connected between the sixth shaft and the first differential gearing for adding to the angular velocity of the second shaft an opposite angular velocity proportional to the angular displacement of the sixth shaft, and means for indicating the angular displacement of the fourth shaft.

4. A mechanical system as in claim 2 including, in addition, a fifth shaft driven through differential gearing from the fourth shaft, a mechanical feedback path including integrating means between the fifth shaft and the last-named differential gearing and means for adding the motion of the fifth shaft to that of the fourth shaft in the feedback path between the fourth shaft and the first-named differential gearing.

RALPH B. BLACKMAN.

REFERENCES CITED The following references are of record in the file of this patent:

UNITED STATES PATENTS Number Name Date 2,071,424 Papello Feb. 28, 1937 2,089,878 Corbin Aug. 10, 1937 2,136,213 Hodgman Nov. 8, 1938 2,206,875 Chaffee July 9, 1940 2,248,072 Fry July 8, 1941 2,377,898 Myers June 12, 1945 2,426,584 Baker SeptA 2, 1947 2,433,006 Weiss Dec. 23, 1947 2,442,792 White June 8, 1948 

